Selecta Mathematica

, Volume 22, Issue 1, pp 1–25 | Cite as

Quantization of line bundles on lagrangian subvarieties

  • Vladimir Baranovsky
  • Victor Ginzburg
  • Dmitry Kaledin
  • Jeremy Pecharich
Article

Abstract

We apply the technique of formal geometry to give a necessary and sufficient condition for a line bundle supported on a smooth Lagrangian subvariety to deform to a sheaf of modules over a fixed deformation quantization of the structure sheaf of an algebraic symplectic variety.

Mathematics Subject Classification

53D55 14D21 

References

  1. 1.
    Beilinson, A., Bernstein, J.: A proof of Jantzen conjectures. I. M. Gelfand Seminar, Adv. Soviet Math., 16(Part 1), 1–50, Amer. Math. Soc., Providence, RI (1993)Google Scholar
  2. 2.
    Beilinson, A., Drinfeld, V.: Quantization of Hitchin’s integrable system and Hecke eigensheaves. http://www.math.uchicago.edu/~mitya/langlands.html
  3. 3.
    Baranovsky, V., Ginzburg, V., Pecharich, J.: Deformation of line bundles on coisotropic subvarieties. Q. J. Math. 63(3), 525–537 (2012)MATHMathSciNetCrossRefGoogle Scholar
  4. 4.
    Bezrukavnikov, R., Kaledin, D.: Fedosov quantization in algebraic context. Mosc. Math. J. 4(3), 559–592, 782 (2004)Google Scholar
  5. 5.
    D’Agnolo, A., Schapira, P.: Quantization of complex Lagrangian submanifolds. Adv. Math. 213(1), 358–379 (2007)MATHMathSciNetCrossRefGoogle Scholar
  6. 6.
    Gabber, O.: The integrability of the characteristic variety. Am. J. Math. 103, 445–468 (1981)MATHMathSciNetCrossRefGoogle Scholar
  7. 7.
    Kashiwara, M.: Quantization of contact manifolds. Publ. Res. Inst. Math. Sci. 32, 1–7 (1996)MATHMathSciNetCrossRefGoogle Scholar
  8. 8.
    Loday, J.-L.: Cyclic homology. Grundlehren der Mathematischen Wissenschaften, 2nd edn, vol. 301. Springer, Berlin (1998)Google Scholar
  9. 9.
    Nest, R., Tsygan, B.: Remarks on modules over deformation quantization algebras. Mosc. Math. J. 4, 911–940 (2004)MATHMathSciNetGoogle Scholar

Copyright information

© Springer Basel 2015

Authors and Affiliations

  • Vladimir Baranovsky
    • 1
  • Victor Ginzburg
    • 2
  • Dmitry Kaledin
    • 3
  • Jeremy Pecharich
    • 4
  1. 1.Department of MathematicsUniversity of California at IrvineIrvineUSA
  2. 2.Department of MathematicsUniversity of ChicagoChicagoUSA
  3. 3.Algebraic Geometry SectionSteklov Mathematical InstituteMoscowRussia
  4. 4.Department of MathematicsPomona CollegeClaremontUSA

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